Adaptive Local Kernels for Lagrangian Modeling of Reactive Transport

Abstract: In recent years, a number of Lagrangian approaches [e.g. 1,2,3,4,5] have been proposed for the numerical solution of nonlinear reactive transport problems. In these approaches, reactions are driven by the interaction between numerical particles that represent either a volume of fluid or a mass of solute. This interaction is a function of the relative position of particles and can be derived mathematically by equipping each particle with a kernel function, sometimes referred to as the smoothing function. In some of these approaches, the choice of the kernel shape and size does not follow objective criteria, but is instead rather heuristic. On the other hand, some authors have proposed that the smoothing function should be derived from the dispersive process [e.g. 2,6]. Recently, Fernàndez-Garcia and Sànchez-Vila [7] proposed that the kernel should represent the probability density distribution of the actual particle location, and should be globally optimized by minimizing the Mean Integrated Squared Error (MISE) of the density estimation. By using the existing algorithms for the determination of a time-dependent globally optimal Kernel Density Estimator (KDE), it is possible to compute probabilities (or rates) of reaction of numerical particles to model simple bimolecular reactions [8] or kinetic reactions of any complexity involving two reactants [9]. Most research on KDE focuses on 1D distributions of limited complexity. However, solute concentration distributions in heterogeneous media exhibit complex features and variations in space and time. For this reason, the globally optimal kernel is often far from being locally optimal, in particular, in those parts of the domain with relative small particle densities, and also in those with particularly abrupt concentration changes (such as mixing fronts). A novel approach is presented to define an adaptive locally optimal kernel function in 1-, 2- or 3-D that is not only time-dependent but also space-dependent. The presented local approaches are compared against the existing global ones, and also against the simple binning method, in the context of a Random Walk Particle Tracking (RWPT) model of reactive transport through a heterogeneous porous medium. The results show that the use of a locally adaptive kernel has a considerably positive impact on the accuracy of concentration estimations and chemical reaction simulations. References: [1] Tartakovsky, A. M., Meakin, P., Scheibe, T. D., & Eichler West, R. M. (2007). Simulations of reactive transport and precipitation with smoothed particle hydrodynamics. Journal of Computational Physics, 222(2), 654–672. [2] Benson, D. A., & Meerschaert, M. M. (2008). Simulation of chemical reaction via particle tracking: Diffusion-limited versus thermodynamic rate-limited regimes. Water Resources Research, 44(12). [3] Herrera, P. A., Massabó, M., & Beckie, R. D. (2009). A meshless method to simulate solute transport in heterogeneous porous media. Advances in Water Resources, 32(3), 413–429. [4] Ding, D., & Benson, D. A. (2015). Simulating biodegradation under mixing-limited conditions using Michaelis-Menten (Monod) kinetic expressions in a particle tracking model. Advances in Water Resources, 76, 109–119. [5] Engdahl, N. B., Benson, D. A., & Bolster, D. (2017). Lagrangian simulation of mixing and reactions in complex geochemical systems. Water Resources Research, 53(4), 3513–3522. [6] Paster, A., Bolster, D., & Benson, D. A. (2013). Particle tracking and the diffusion-reaction equation. Water Resources Research, 49(1), 1–6. [7] Fernàndez-Garcia, D., & Sanchez-Vila, X. (2011). Optimal reconstruction of concentrations, gradients and reaction rates from particle distributions. Journal of Contaminant Hydrology, 120–121(C), 99–114. [8] Rahbaralam, M., Fernàndez-Garcia, D., & Sanchez-Vila, X. (2015). Do we really need a large number of particles to simulate bimolecular reactive transport with random walk methods? A kernel density estimation approach. Journal of Computational Physics, 303, 95–104. [9] Sole-Mari, G., Fernàndez-Garcia, D., Rodríguez-Escales, P., & Sanchez-Vila, X. (2017). A KDE-Based Random Walk Method for Modeling Reactive Transport with Complex Kinetics in Porous Media. Water Resources Research.